^{1,a)}, Karthikeyan Rajendran

^{2}and Ioannis G. Kevrekidis

^{3}

### Abstract

We consider the simplest network of coupled non-identical phase oscillators capable of displaying a “chimera” state (namely, two subnetworks with strong coupling within the subnetworks and weaker coupling between them) and systematically investigate the effects of gradually removing connections within the network, in a random but systematically specified way. We average over ensembles of networks with the same random connectivity but different intrinsic oscillator frequencies and derive ordinary differential equations (ODEs), whose fixed points describe a typical chimera state in a representative network of phase oscillators. Following these fixed points as parameters are varied we find that chimera states are quite sensitive to such random removals of connections, and that oscillations of chimera states can be either created or suppressed in apparent bifurcation points, depending on exactly how the connections are gradually removed.

Chimera states are known to occur in networks of identical phase oscillators and are characterised by some fraction of the oscillators being synchronised, while the remainder are asynchronous. Previous studies of these states have considered highly symmetric networks, with all-to-all connectivity. The question as to the robustness of these states with respect to changes in the network structure naturally arises. Here, we systematically investigate this issue for what is arguably the simplest network that shows chimera states, considering two different systematic ways of perturbing an all-to-all connected network. By varying parameters controlling the oscillators’ dynamics, as well as parameters controlling the topology of the network, we can determine the effects of changing this topology on the existence and stability of chimera states. We find that chimera states are quite sensitive to changes in the network topology, and that one of the perturbations considered suppresses oscillations of chimera states, while the other promotes oscillations.

The work of K.R. and I.G.K. was partially supported by the AFOSR and the US Department of Energy. C.R.L. was supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.

I. INTRODUCTION

II. MODEL AND ANALYSIS

III. RESULTS

A. All-to-all coupling

B. “Sparse” coupling matrices

1. Erdös-Rényi type networks

2. Chung-Lu-type networks

IV. DISCUSSION

### Key Topics

- Oscillators
- 61.0
- Networks
- 52.0
- Bifurcations
- 26.0
- Coupled oscillators
- 12.0
- Network topology
- 6.0

## Figures

(Color online) A chimera solution of Eqs. (1) and (2) for full connectivity matrices. Top: a temporal snapshot of the phase as a function of index, *i*, where those in population 1 are numbered 1 to 500, and those in population 2 are numbered 501 to 1000. Bottom left: histogram of oscillator phases in population 1. Bottom right: histogram of oscillator phases in population 2. Parameter values: . All chosen from a Lorentzian distribution with half-width-at-half-maximum 0.001.

(Color online) A chimera solution of Eqs. (1) and (2) for full connectivity matrices. Top: a temporal snapshot of the phase as a function of index, *i*, where those in population 1 are numbered 1 to 500, and those in population 2 are numbered 501 to 1000. Bottom left: histogram of oscillator phases in population 1. Bottom right: histogram of oscillator phases in population 2. Parameter values: . All chosen from a Lorentzian distribution with half-width-at-half-maximum 0.001.

(Color online) Steady states of Eqs. (29) and (30) , where . This figure relates to the all-to-all coupled case in the limit . Lines indicate symmetric solutions for which ; solid blue: stable, dashed red: unstable. Points indicate chimera states where ; blue: stable, red: unstable. The symmetric state undergoes two pitchfork bifurcations as is varied. The lower panel is an enlargement of the upper panel. Parameters: .

(Color online) Steady states of Eqs. (29) and (30) , where . This figure relates to the all-to-all coupled case in the limit . Lines indicate symmetric solutions for which ; solid blue: stable, dashed red: unstable. Points indicate chimera states where ; blue: stable, red: unstable. The symmetric state undergoes two pitchfork bifurcations as is varied. The lower panel is an enlargement of the upper panel. Parameters: .

(Color online) Bifurcations of fixed points of Eqs. (29) and (30) . Green dots: Hopf bifurcation of the stable chimera; blue line: saddle-node bifurcation of the stable and unstable chimera; red dotted: pitchfork bifurcation of the symmetric state in which . A stable chimera exists in the region bounded by the left-most pitchfork bifurcation, the saddle-node bifurcation and the Hopf bifurcation. Figure 2 is a horizontal “slice” through this figure at *E* = 0.2. Parameter: .

(Color online) Bifurcations of fixed points of Eqs. (29) and (30) . Green dots: Hopf bifurcation of the stable chimera; blue line: saddle-node bifurcation of the stable and unstable chimera; red dotted: pitchfork bifurcation of the symmetric state in which . A stable chimera exists in the region bounded by the left-most pitchfork bifurcation, the saddle-node bifurcation and the Hopf bifurcation. Figure 2 is a horizontal “slice” through this figure at *E* = 0.2. Parameter: .

(Color online) Hopf (dashed) and saddle-node and left-most pitchfork (solid) bifurcations of fixed points of Eqs. (29) and (30) for (black, also shown in Fig. 3 ), (blue) and (red). The arrows indicate how the curves move as is increased. A stable chimera exists within the region bounded by curves of the same colour, for the corresponding value of .

(Color online) Hopf (dashed) and saddle-node and left-most pitchfork (solid) bifurcations of fixed points of Eqs. (29) and (30) for (black, also shown in Fig. 3 ), (blue) and (red). The arrows indicate how the curves move as is increased. A stable chimera exists within the region bounded by curves of the same colour, for the corresponding value of .

(Color online) Degree distributions for Erdös-Rényi-type networks (top row) and Chung-Lu-type networks (bottom row). See text for explanation of parameters. Here *N* = 300.

(Color online) Degree distributions for Erdös-Rényi-type networks (top row) and Chung-Lu-type networks (bottom row). See text for explanation of parameters. Here *N* = 300.

(Color online) A chimera solution in a network with Erdös-Rényi-type connectivity. Oscillators in population 1 are numbered 1 to 300, those in population 2 are numbered 301 to 600. Top and middle panels show fixed points of Eqs. (31) and (32) , while the bottom panel is a snapshot of Eqs. (1) and (2) for the same connectivity matrices. Top: magnitudes of the (left) and (right). Middle: argument of the (left) and (right). Other parameters: .

(Color online) A chimera solution in a network with Erdös-Rényi-type connectivity. Oscillators in population 1 are numbered 1 to 300, those in population 2 are numbered 301 to 600. Top and middle panels show fixed points of Eqs. (31) and (32) , while the bottom panel is a snapshot of Eqs. (1) and (2) for the same connectivity matrices. Top: magnitudes of the (left) and (right). Middle: argument of the (left) and (right). Other parameters: .

The values of for which there is a saddle-node bifurcation of fixed points of Eqs. (25), (26), (27), (28), and (31) and (32) for Erdös-Rényi-type random matrices *A, B, C*, and *D*, as a function of *p*. At each value of *p*, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 are shown. Other parameters: .

The values of for which there is a saddle-node bifurcation of fixed points of Eqs. (25), (26), (27), (28), and (31) and (32) for Erdös-Rényi-type random matrices *A, B, C*, and *D*, as a function of *p*. At each value of *p*, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 are shown. Other parameters: .

Values of *E* at which a Hopf bifurcation of a stable stationary chimera state occurs for an Erdös-Rényi-type network, for different values of *p*. At each value of *p*, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 values of *E* are shown. Other parameters: .

Values of *E* at which a Hopf bifurcation of a stable stationary chimera state occurs for an Erdös-Rényi-type network, for different values of *p*. At each value of *p*, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 values of *E* are shown. Other parameters: .

(Color online) Onset of a Hopf bifurcation caused by randomly removing (with equal probability) a few of the connections within the phase oscillator network (1) and (2) . At *t* = 1000 we switched the connectivity matrices *A, B, C*, and *D* from full to Erdös-Rényi-type matrices with *p* = 0.9. We define and . Other parameters: .

(Color online) Onset of a Hopf bifurcation caused by randomly removing (with equal probability) a few of the connections within the phase oscillator network (1) and (2) . At *t* = 1000 we switched the connectivity matrices *A, B, C*, and *D* from full to Erdös-Rényi-type matrices with *p* = 0.9. We define and . Other parameters: .

The values of for which there is a saddle-node bifurcation of fixed points of Eqs. (25), (26), (27), (28), and (31) , (32) for Chung-Lu-type random matrices *A, B, C,* and *D*, as a function of *r*. At each value of *r*, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 are shown. Other parameters: .

The values of for which there is a saddle-node bifurcation of fixed points of Eqs. (25), (26), (27), (28), and (31) , (32) for Chung-Lu-type random matrices *A, B, C,* and *D*, as a function of *r*. At each value of *r*, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 are shown. Other parameters: .

Values of *E* at which a Hopf bifurcation of a stable stationary chimera state occurs for a Chung-Lu-type network, for different values of *r*. At each value of *r*, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 values of *E* are shown. Other parameters: .

Values of *E* at which a Hopf bifurcation of a stable stationary chimera state occurs for a Chung-Lu-type network, for different values of *r*. At each value of *r*, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 values of *E* are shown. Other parameters: .

(Color online) Suppression of oscillations by removing some of the connections within the oscillator network (1) and (2) . At *t* = 1000 we switched the connectivity matrices *A, B, C*, and *D* from full to Chung-Lu-type matrices with *r* = 0.1. We define and . Other parameters: .

(Color online) Suppression of oscillations by removing some of the connections within the oscillator network (1) and (2) . At *t* = 1000 we switched the connectivity matrices *A, B, C*, and *D* from full to Chung-Lu-type matrices with *r* = 0.1. We define and . Other parameters: .

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